Comparing apples and oranges: fold-change detection of
multiple simultaneous inputs – Supporting Information
Material S1
Yuval Hart, Avraham E. Mayo, Oren Shoval and Uri Alon
Dept. Molecular Cell biology, Weizmann Institute of Science, Rehovot, Israel
Contents
A MWC model for interacting ligands binding a shared receptor integrates signals in a log-linear
manner ............................................................................................................................................. 1
A MWC model for exclusive binding of ligands to a shared receptor cannot hold FCD ................ 3
A system with two inputs, two internal layer components and an output with a FCD response for
each input enables mapping of the two internal layer components output integration by studying
the equivalent fold response to the two signal folds ........................................................................ 3
A MWC model for interacting ligands binding a shared receptor integrates signals
in a log-linear manner
In the main text, we have assumed that binding of ligands to receptor is non-interacting.
Here, we wish to relax this assumption and allow the two binding events to affect one
another.
In order to analyze the case of interacting binding events, we use Tu et al. energy
function formulation [1] of the MWC model [2]. In this formulation, the receptor has N1
subunits for binding ligand 1 and N2 subunits for binding ligand 2. Ligand occupancy of
each subunit is denoted by 𝜎𝑖1 for ligand 1 and 𝜎𝑖2 for ligand 2 and can equal either zero
or one: 𝜎𝑖1 , 𝜎𝑖2 = {0,1}. In the concerted model, receptor activity s is either active (s=1)
or inactive (s=0). Thus, the energy of the receptor depends on the variables s and 𝜎{1,2},𝑖
as follows:
(S1)
𝐻 = 𝑠 (𝐻0 + 𝜖1 Σ𝑖1 𝜎𝑖1 + 𝜖2 Σ𝑖2 𝜎𝑖2 + 𝜖3 (Σ𝑖1 𝜎𝑖1 )(Σ𝑖2 𝜎𝑖2 )) + 𝜇1 Σ𝑖 𝜎𝑖1 +
𝜇2 Σ𝑖 𝜎𝑖2 + 𝜇3 (Σ𝑖 𝜎𝑖1 )(Σ𝑖 𝜎𝑖2 )
Where 𝐻0 is the energy difference between the active and inactive state in the absence of
ligands. Each occupied receptor subunit suppresses receptor activity by increasing the
energy of the active state by 𝜖1 or 𝜖2 . The joint effect of both ligands occupation on the
activity is described by the energy cost 𝜖3 . The energies 𝜇1 , 𝜇2 are the energies for ligand
binding for the inactive state and depends on ligand concentrations and the dissociation
constants of ligand to the inactive state, 𝐾𝑖,{1,2} . 𝜇3 is the energy cost of having both
ligand subunits occupied, and describes the interaction between ligand binding. The
relation between energy parameters and the MWC model are:
(S2)
𝑒 −𝐻0 = 𝐾𝑒𝑞 , 𝑒 −𝜖1 = 𝐶1 ,
(S3)
𝑒 −𝜇1 = 𝐾 1 , 𝑒 −𝜇2 = 𝐾 2 , 𝑒 −𝜇3 = 𝐾1
𝐿
𝑒 −𝜖2 = 𝐶2 ,
𝐿
𝑖,1
𝑖,2
𝑒 −𝜖3 = 𝐶12
𝐿 𝐿2
𝑖,12
Where 𝐾𝑒𝑞 is the equilibrium constant, L1 and L2 are the concentrations of ligand 1 and
ligand 2 and the dissociation constants for the active state are given by 𝐾𝑎,{1,2} =
𝐾𝑖,{1,2}
𝐶{1,2}
.
To calculate the average receptor activity, a= <s>, one needs to calculate the partition
function 𝑍 = Σ{𝑎𝑙𝑙 𝑠𝑡𝑎𝑡𝑒𝑠} 𝑒 −𝐻 and then differentiate its logarithm with respect to H0
(S4)
1 𝜕𝑍
𝑎 =< 𝑠 >= − 𝑍 𝜕𝐻 =
0
𝑁1
𝑁2
𝑁1+𝑁2
𝐶 𝐿
𝐶 𝐿
𝐶 𝐿 𝐿
𝐾𝑒𝑞 (1+ 1 1 ) (1+ 2 2 ) (1+ 12 1 2 )
𝐾𝑖,1
𝐾𝑖,2
𝐾𝑖,12
𝑁1
𝑁2
𝑁1 +𝑁2
𝑁
𝑁2
𝑁1+𝑁2
𝐿1
𝐿2
𝐿1 𝐿2
𝐶 1 𝐿1 1
𝐶 𝐿
𝐶 𝐿 𝐿
(1+
) (1+
) (1+
)
) (1+ 2 2 ) (1+ 12 1 2 )
+𝐾𝑒𝑞 (1+
𝐾𝑖,1
𝐾𝑖,2
𝐾𝑖,12
𝐾𝑖,1
𝐾𝑖,2
𝐾𝑖,12
Note that the case of no interactions between the two ligand binding events is achieved
𝐿 𝐿2
when 𝜖3 = 𝜇3 = 0 which is equivalent to 𝐶12 = 1 and 𝐾1
𝑖,12
= 1 such that Eq.(S4) results
in the known MWC formula for two ligands.
Generally, receptor activity does not follow a power-law behavior that will allow for a
fold-change detection response. However, in the case where 𝐾𝑒𝑞 is small compared with
one, and ligand concentrations are within a range, such that they hold: 𝐾𝑖,12 ≪ 𝐿1 𝐿2 ≪
𝐾𝑖,12
𝐶12
then power-law behavior is achieved for two cases: One, when ligands
concentrations are within the range 𝐾𝑖,1 ≪ 𝐿1 ≪
𝐾𝑖,1
𝐶1
and 𝐾𝑖,2 ≪ 𝐿2 ≪
𝐾𝑖,2
𝐶2
. In this case the
receptor activity is given by:
(S5)
𝑎=
𝐾𝑒𝑞
𝑁1
𝐿
( 1)
𝐾𝑖,1
𝑁2
𝑁1 +𝑁2
𝐿
𝐿 𝐿
( 2 ) ( 1 2)
𝐾𝑖,2
𝐾𝑖,12
In a sense, this case resembles the case of non-interacting binding because the above
conditions are met when 𝐾𝑖,1 𝐾𝑖,2 ≃ 𝐾𝑖,12 and 𝐶1 𝐶2 ≃ 𝐶12 which is met when two ligand
binding is the multiplication of single binding probabilities.
The second case occurs when ligand concentrations are very low, meaning when 𝐿1 ≪
𝐾𝑖,1 and 𝐿2 ≪ 𝐾𝑖,2 but yet 𝐾𝑖,12 ≪ 𝐿1 𝐿2 ≪
(S6)
𝑎 = 𝐾𝑒𝑞
𝐾𝑖,12
𝐶12
such that the receptor activity is given by:
1
𝑁1 +𝑁2
𝐿 𝐿
( 1 2)
𝐾𝑖,12
This condition is met when 𝐾𝑖,12 ≪ 𝐾𝑖,1 𝐾𝑖,2 which means the two ligands are synergistic
in the sense that binding of the two ligands together is highly favorable compared to
binding of each one of them by itself.
In both of these cases, the FCD demands are met and the log-linear multiplicative nature
of the ligands is kept.
A MWC model for exclusive binding of ligands to a shared receptor cannot hold
FCD
For the case of exclusive binding, one can use the same Hamiltonian as used for the noninteracting ligands, but count only states which are exclusive, meaning that only ligand 1
or ligand 2 bind the receptor, but not together. The resulting receptor activity is given by:
𝑁1
𝑁2
𝐶 𝐿
𝐶 𝐿
𝐾𝑒𝑞 ((1+ 1 1 ) +(1+ 2 2 ) )
(S7)
𝑎=
𝑁1
𝐿
(1+ 1 )
𝐾𝑖,1
𝐾𝑖,1
𝑁2
𝐿
+(1+ 2 )
𝐾𝑖,2
𝐾𝑖,2
𝑁1
𝑁2
𝐶 𝐿
𝐶 𝐿
+𝐾𝑒𝑞 ((1+ 1 1 ) +(1+ 2 2 ) )
𝐾𝑖,1
𝐾𝑖,2
Here, since the two ligands are summed rather than multiplied, there is no regime where
the receptor activity can be represented by a power-law of both ligands. The only case for
a possible FCD regime is the trivial case where one ligand concentration is extremely
small and then FCD can be attained for the second ligand in its FCD range (𝐾𝑖,1 ≪ 𝐿1 ≪
𝐾𝑖,1
𝐶1
).
A system with two inputs, two internal layer components and an output with a FCD
response for each input enables mapping of the two internal layer components
output integration by studying the equivalent fold response to the two signal folds
In this section we describe a specific setting which could yield a mapping between the
internal components integration scheme and the fold response to the two signals:
Consider a case where the two internal layer components have the same time scales and
in addition the system shows exact adaptation in both internal layer components.
In this case, the two internal layer components, x1 and x2, have an explicit integral
feedback form [3–5] on the output y. This means that one can write their dynamic
equations in the following form
(S8)
𝑥1̇ = 𝐺1 (𝑦 − 𝑦0 )𝑓1 (𝑢1 , 𝑥1 )
(S9)
𝑥2̇ = 𝐺2 (𝑦 − 𝑦0 )𝑓2 (𝑢2 , 𝑥2 )
where 𝐺1 (𝑦 − 𝑦0 ) has a zero at 𝑦0 , the same zero as 𝐺2 (𝑦 − 𝑦0 ) in order for the system
to reach the exact same stationary output level, 𝑦0 . Then, for y close to the steady-state
value, 𝑦0 , the dynamic equations (S8-S9) have a conserved quantity, 𝑥1̇ 𝑓2 (𝑢2 , 𝑥2 ) −
𝑥2̇ 𝑓1 (𝑢1 , 𝑥1 ) = 0. Thus, we get the following implicit equation:
𝑥1̇
(S10)
𝑓1 (𝑢1 ,𝑥1 )
= 𝑓 (𝑢 2̇ ,𝑥
𝑥
2
2
2)
In turn, this leads to a mapping of x1 onto x2, 𝑥1 = 𝑄(𝑥2 ) and so the two internal layer
variables are coupled by the integral feedback mechanism.
The equivalent response of one ligand to a fold change in both ligands manifests the
integration scheme of the internal layer components (x1,x2) to the output of the system
𝑢
𝑢
𝑢
𝑢
(y). The dynamic equation of y holds: 𝑦̇ = 𝑔 (𝑥1 , 𝑥2 , 𝑦) = 𝑔 (𝑄(𝑥1 ) , 𝑥2 , 𝑦) =
1
𝑢
2
2
2
𝑢
2
𝑔 (𝑥 1 , 𝑄−1 (𝑥
, 𝑦). Hence, one can map the 'input function' of the output y(t) by mapping
)
1
1
the output's response to different folds.
Therefore, an FCD system with an explicit exact adaptation and no delays in both inputs
can be internally mapped by studying the invariant response for different fold changes in
the two ligands.
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